The Time Between: Continuously-defined accessibility functions for schedule-based transportation systems

Accessibility is traditionally considered to be a property of a point or region in space, and to be invariant over time (or at least over some computationally convenient time interval). However, a location’s accessibility can vary over time on a wide range of scales. This temporal variation is especially significant for schedule-based transportation systems. Current integral measures of accessibility generally reflect the accessibility only at points in time corresponding to the departures of one or more trips; accessibility between these time points remains unconsidered and undefined. Consequently, these measures are insensitive to changes in route frequency and the distribution of trip departure times. Furthermore, these approaches ignore the disutility experienced by a system user who is limited to departing or arriving at scheduled times rather than at preferred times. As a result, they systematically overestimate the accessibility experienced by users of scheduled transportation systems. This paper establishes new methods for representing the accessibility provided by a schedule-based transportation system from a specific location as a continuously-defined accessibility function (CDAF) of desired departure time, defined for all time points. Using schedule and route information from metropolitan transit providers, applications of these methods are demonstrated to gain new insight into the accessibility provided by real-world transportation systems. Four examples are developed to represent common service types in metropolitan transit networks. The results confirm that accessibility is significantly overestimated by measuring single points and show that trip frequency is more valuable for sustained accessibility than high accessibility on individual trips. Owen, Anderson, and Levinson 2 INTRODUCTION Accessibility, the ease of reaching destinations, is traditionally considered to be a property of a point or region in space, and to be invariant over time (or at least over some computationally convenient time interval). The concept has been well-described in the literature, and there are numerous definitions (1, 2, 3, 4, 5, 6, 7, 8). However, a location’s accessibility can vary over time on a wide range of scales. There are many works on space-time accessibility in the literature, beginning with Hägerstrand (9), who advanced the idea of space-time prisms to represent an individual’s accessibility. His theory incorporated the idea that activities occur at specific locations for limited time periods. Kwan (10) drew a distinction between space-time measures, which follow individuals throughout the day, and integral (or place-based) measures and concluded that the two types are independent of each other. Miller (11) looked at applications of Hägerstrand’s space-time prisms, and incorporated users’ preferred start and end times and overall time budget. This brought the theory closer to the user experience, as commuters in particular frequently have a fixed arrival time in the morning peak and a fixed departure time in the afternoon peak. In a later work, Miller (12) discussed the idea of fixed and flexible activities. Building on this idea, users going to and from fixed activities are more sensitive to the temporal constraints imposed by scheduled transportation services. Weber and Kwan (13) concluded that temporal effects from both the time-dependence of automobile travel times and from business hours have significant effects on accessibility. Temporal variation for schedule-based transportation systems is even more significant than for car travel because it becomes an issue of service availability in addition to speed. Current integral measures of accessibility generally reflect the accessibility only at points in time corresponding to the departures of one or more trips; accessibility between these time points remains unconsidered and undefined. Consequently, these measures are insensitive to changes in route frequency and the distribution of trip departure times. Furthermore, these approaches ignore the disutility experienced by a system user who is limited to departing or arriving at scheduled times rather than at preferred times. As a result, they systematically overestimate the accessibility experienced by users of scheduled transportation systems. One work that looks at this issue is Kim and Kwan (14), which shows that space-time accessibility measures that trivialize the issues of schedule availability and network topology exhibit this problem of overestimation. This paper establishes newmethods for representing the accessibility provided by a schedulebased transportation system from a specific location as a continuously-defined accessibility function (CDAF) of desired departure time, defined for all time points. Using schedule and route information from metropolitan transit providers, the application of these methods is demonstrated to gain new insight into the accessibility provided by real-world transportation systems. Here, the aim is to explore whether trip frequency or trip accessibility is more important to a sustained transit accessibility over time. While traditional measures of accessibility assume that users will adjust their departure times to meet the schedule provided by a transportation system, a central benefit of a CDAF-based approach to accessibility is that accessibility information is available for any desired departure time. The CDAF enables detailed investigation of how accessibility at a given location varies over time, and how it is influenced by the time distribution and properties of trips departing from that location. When trip schedule and route information is available, it becomes possible to plot accessibility over a time period of interest and compute statistical properties such as mean and median accessibility, both of which can be calculated for any stop on a transit network. The range Owen, Anderson, and Levinson 3 and variation of accessibility during a time period can also be calculated. These measures provide a better representation of the accessibility experienced by users. Several examples are presented as proof of concept, and the relationships are investigated between accessibility measures provided using a CDAF to traditional time-point based accessibility measures. METHODOLOGY Construction of the CDAF for a specific location begins with the identification of all trip departures in a time period of interest. For each trip departure, the departure time and a vector T representing the travel times provided by that trip to all reachable destinations are retained. A vector O is also established which provides the number of opportunities at each reachable destination: T =< tt1, tt2, ..., ttn > O =< o1, o2, ..., on > Given these sets of travel times and opportunity counts, it is possible to implement a wide range of accessibility metrics. Most contemporary implementations can be traced at least back to (15), who proposes a measure where potential destinations are weighted by a function of their access cost and then summed: